The syntonic tuning continuum (Milne 2007).
Regular temperament is any tempered system of musical tuning such that each frequency ratio is obtainable as a product of powers of a finite number of generators, or generating frequency ratios. The classic example of a regular temperament is meantone temperament, where the generating intervals are usually given in terms of a slightly flattened fifth and the octave.
The bestknown example of a linear temperaments is meantone, but others include the schismatic temperament of Hermann von Helmholtz and miracle temperament.
Mathematical description
If the generators are all of the prime numbers up to a given prime p, we have what is called plimit just intonation. Sometimes some irrational number close to one of these primes is substituted (an example of tempering) to favour other primes, as in twelve tone equal temperament where 3 is tempered to 2^{19/12} to favour 2, or in quartercomma meantone where 3 is tempered to 2·5^{1/4} to favor 2 and 5.
In mathematical terminology, the products of these generators define a free abelian group. The number of independent generators is the rank of an abelian group. The rankone tuning systems are equal temperaments, all of which can be spanned with only a single generator. A ranktwo temperament has two generators. Hence, meantone is a rank2 temperament.
In studying regular temperaments, it can be useful to regard the temperament as having a map from plimit just intonation (for some prime p) to the set of tempered intervals. To properly classify a temperament's dimensionality one must determine how many of the given generators are independent, because its description may contain redundancies. Another way of considering this problem is that the rank of a temperament should be the rank of its image under this map.
For instance, a harpsichord tuner it might think of quartercomma meantone tuning as having three generators—the octave, the just major third (5/4) and the quartercomma tempered fifth—but because four consecutive tempered fifths produces a just major third, the major third is redundant, reducing it to a ranktwo temperament.
Other methods of linear and multilinear algebra can be applied to the map. For instance, a map's kernel (otherwise known as "nullspace") consists of plimit intervals called commas, which are a property useful in describing temperaments.
External links

Xenharmonic wiki, Regular Temperaments

A. Milne, W. A. Sethares, and J. Plamondon, Isomorphic Controllers and Dynamic Tuning— Invariant Fingering Over a Tuning Continuum, Computer Music Journal, Winter 2007

Microtonal scales: Rank2 2step (MOS) scalesHolmes, Rich,

Regular TemperamentsSmith, Gene Ward,

Barbieri, Patrizio. Enharmonic instruments and music, 14701900. (2008) Latina, Il Levante Libreria Editrice
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